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I am reading the paper Certifying almost all quantum states with few single-qubit measurements. The main result of the paper (Theorem 1) is that given an $n$-qubit target pure state $|\psi\rangle$ and many copies of $\rho$, by performing single-qubit Pauli measurements on different copies of $\rho$, it can be certified whether $\rho$ is close to (using fidelity) $|\psi\rangle\langle\psi|$ or not with high probability.

The thing I don't understand is why not just do the self-test procedure to certify that? Since All Pure Bipartite Entangled States can be Self-Tested, we can simulate two players using optimal strategies and certify whether $\rho$ is close to $|\psi\rangle\langle \psi|$ from the winning rate. Since we can exactly know the strategy the players use (instead of up to local rotations), we can know the quantum states exactly (instead of up to local rotations). So what is the advantage of the new method in this paper?

qmww987
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1 Answers1

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Indeed bipartite pure quantum states can be certified using self-testing, but this may not be the most practical nor efficient way to gain information about an unknown quantum state in many settings. In particular, you must remember that the context for self-testing is the device-independent paradigm, where the quantum devices are untrusted. This is distinguished from quantum tomography where we are provided with a quantum state (or many copies) and want to learn something about it. Here are some more distinctions:

  • The self-testing protocol in [2] requires two non-communicating quantum devices that share the target quantum state. This is already much more complicated than the setting of [1] where just a single quantum device is required. Not to mention, the issues with the non-communicating assumption itself.
  • To inspect the correlation from the self-testing scenario in [2] one needs to perform multiple rounds to get an accurate description of the correlation. So there isn't an advantage to self-testing over the need for multiple copies of the state in [1].
  • The protocol in [2] works only for bipartite pure states, while the protocol in [1] does not have this caveat.
  • Unlike in [1] the complexity of the measurements required in [2] is not clear. Although general self-testing typically allows one to effectively certify the measurements in addition to the state, I believe the protocol in [2] lacks this property.
  • Although the procedure in [1] is superior in most contexts, the advantage of the procedure in [2] is the device-independent nature of the certification protocol.
  • Lastly, a self-test only certifies a state up to local isometries it does not exactly imply that the target state and the employed state are close in norm (or fidelity).

[1] Certifying almost all quantum states with few single-qubit measurements, https://arxiv.org/pdf/2404.07281

[2] All Pure Bipartite Entangled States can be Self-Tested, https://arxiv.org/pdf/1611.08062

Condo
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